UK Nonlinear News, February 2001

Ergodic Theory and Complex Dynamics Meeting 2001

Omri Sarig

This was a 3 days meeting which was organized by the Dynamics Research Group in the Mathematics Department of the University of Warwick, UK from 19/1/2001 to 21/1/2001. The meeting was organized by A. Epstein, G. Havard, O. Sarig, and S. Van-Strien from Warwick, and was sponsored by the European Scientific Foundation as part of the PRODYN program. 

List of Talks and Abstracts: 

Prof. Jon Aaronson  
Sums without Maxima (joint work with H. Nakada) 
Sums of positive random variables with "barely infinite" expectations (excluding maximal values) sometimes enjoy a SLLN. There is an application to some non uniformly hyperbolic ("intermittent") interval maps (joint work with Hitoshi Nakada).

Nalini Anantharaman  
Counting geodesics optimal in homology

Artur Avila de Melo
Statistical properties of unimodal maps (Joint work with Carlos Gustavo Moreira) 
We discuss recent developments on the description of the quadratic family and other families of unimodal maps from the measure-theoretical point of view. In particular, we show that typical unimodal maps are either hyperbolic or Collet-Eckmann, with polynomial recurrence of the critical orbit. In both cases the dynamics is very nice from the statistical point of view.

Viviane Baladi
Floquet spectrum of weakly coupled analytic maps (Joint work with HH Rugh) 
We consider translation invariant weakly coupled analytic expanding circle maps on Z^D, with small coupling strength and study the spectrum of the associated transfer operator restricted to the eigenspaces of the spatial translation operators. (The operators commute.) We exhibit smooth curves of eigenspaces and eigenfunctions (as functions of the crystal momenta of the spatial translations). 

Xavier Bressaud
Expanding interval maps with intermittent-like behaviour: Physical measures and scales of time

Arnaud Cheritat
Virtual Siegel disks
Local connectivity and measure zero results for Julia sets of some maps z -> rho.z + z^2 having a Siegel disk at 0, have been proved with the use of a model, a Blaschke fraction, usually given by a formula. This fraction can also be constructed in a geometric way. By analogy, one can construct geometrically some models (for which one has no formula), that enable to adapt the results to virtual Siegel disks of Julia-Lavaurs sets that appear in parabolic implosion. 

 Zaqueu Coelho 
The loss of tightness of time distributions for rotations
For an irrational rotation of the circle, the distribution of times to hit a small interval (when rescaled by the size of the interval) may lose tightness when the interval shrinks to a point. 

Sebastien Ferenczi
Structure of three-interval exchange transformations (joint work with C. Holton and L. Zamboni) 
We present a detailed study of the combinatorial, spectral and ergodic properties of three-interval exchange transformations. We give first a necessary and sufficient condition for a symbolic system to be a natural coding of a three-interval exchange, answering an old question of Rauzy; this allows us to define a new algorithm of simultaneous approximations for two real numbers, generalizing the well-known interaction between rotations, Sturmian sequences and the usual continued fraction approximation. Then we define a recursive method of generating three sequences of nested Rokhlin stacks which describe the system from a measure-theoretic point of view and which in turn gives an explicit characterization of the eigenvalues. We obtain necessary and sufficient conditions for weak mixing which, in addition to unifying all previously known examples, allow us to exhibit new interesting examples of weakly mixing three-interval exchanges. Finally we give affirmative answers to two long standing questions posed by W.A. Veech on the existence of three-interval exchanges having irrational eigenvalues and discrete spectrum. 

Oliver Jenkinson
Cohomology classes for Dynamically Non-Negative Functions" (joint work with T. Bousch) 
Given a dynamical system $T:X \to X$, we say that a continuous function $f:X\to R$ is dynamically non-negative (resp. dynamically positive) if $\int f dm \ge 0$ (resp. $\int f dm > 0$ ) for every $T$-invariant probability measure $m$. Given such a function $f$, can we always find a bona fide non-negative (resp. positive) function $g$ in the same dynamic cohomology class? (i.e. $f-g= uT - u$ for some continuous $u$).   If $f$ has some regularity (eg Holder, $C^k$, analytic, etc), can we choose this $g$ to have the same regularity? We consider these problems in the case where $T$ is an expanding map. 

Marc Kesseboehmer
Poisson limit laws for the Gauss map
We introduce a general theorem which states sufficient conditions for schemes of dependent random variables to fulfill a Poisson limit law. Also explicit bounds for the error terms are provided. We then apply this theorem to the return process for the (measure theoretical) dynamical system given by the Gauss map $\sigma$: For almost every $x\in [0,1]$ we have \[ \mu \{x\in [0,1]: \sum _{i=1}^{[\lambda / \mu(B_r (x))]} \chi _{B_r(x)}\circ \sigma ^i (x)=k\}\to \exp(-\lambda)\lambda^k /k\! \] with exponential rate ($\mu$ denotes the Gauss measure). 

Makoto Mori
Low discrepancy sequences generated by piecewise linear transformations
We construct a uniformly distributed sequences by expanding maps, and characterizes discrepancy of sequences by the spectra of the Perron-Frobenius operator. 

Duncan Sands
Metric attractors for smooth unimodal maps (joint work with J. Graczyk & G. Swiatek)
We prove decay of geometry for $C^3$ unimodal maps with $C^3$ non-flat critical points. As a consequence we obtain the classification of the measure theoretical attractors for $C^3$ unimodal maps with non-flat critical points of order 2. 

Richard Sharp
A Local limit theorem for closed geodesic and homology
We discuss the distribution of closed geodesics on a negatively curved manifold. We study the asymptotics of the number of closed geodesics lying in a prescribed homology class. Under certain conditions, we obtain a local limit theorem which gives information which is uniform as the homology class varies. 

Michel Zinsmeister
Continuity with respect to the phase of Hausdorff dimension of Julia-Lavaurs sets

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Last Updated: 5th February 2001.